The Information Manifold for Relatively Bounded Potentials

by R. F. Streater

Proc. Steklov Inst. Math., 228, 217-235, 2000

ABSTRACT

We construct a Banach manifold of states, which are Gibbs states for potentials that are form-bounded relative to the free Hamiltonian. We construct the (+1)-affine structure and the (+1)-connection.

Keywords: information manifold, Fisher metric, quantum geometry, Bogoliubov metric, Kubo theory, statistical manifold.

NOTES

Information geometry is a very elegant way to study the theory of estimation in statistics; the subject has since been introduced into the study of thermodynamics, and also neural nets and finance. It arose from the important work of R. A. Fisher in the 1920's, who developed statistics into a mathematical discipline. He invented the concept of "information", and showed that the variance of any estimator cannot be smaller than the inverse of his information. The mathematical statisticians Rao and Cramer expressed the idea of an information manifold in terms of Riemannian geometry in 1947-48; the Fisher information then reappears as the Riemannian metric of the manifold. Related ideas appeared in physics in 1957 in the hands of E. T. Jaynes, and in chemistry soon after. This work concerned equilibrium states, also known as Gibbs states. Ingarden developed a theory of non-equilibrium statistical mechanics from 1961, using the information manifold as the collection of macro-states of the system. I have developed this programme into a self-contained theory of non-equilibrium thermodynamics, first as a paper, and then as a book Statistical Dynamics. The quantum version has been introduced by Chentsov, and developed by Petz. Until the work of Pistone and Sempi of 1995, all results were rigorously established only for systems of finitely many degrees of freedom. The problem is, the theory of infinite dimensional manifolds has not been developed as much as the finite-dimensional case. Pistone and Sempi actually construct a manifold modelled on the Banach space of Orlicz functions. This work has been applied to some problems in mathematical finance by Fukumizu. There remained the problem of setting up the manifold in the infinite-dimensional quantum case. The present paper is a small start; small because a limited class of states is considered, and only one of the two important affine structures is constructed.

The paper, including the MAPLE program, can be seen on the archives math-ph/9910035

This is part of an ongoing programme with H. Nencka, G. Burdet, P. Combe, G. Pistone and P. Gibilisco to construct the information manifold for a quantum system. For later work see my list of publications.